scholarly journals Fourth order q-difference equation for the first associated of the q-classical orthogonal polynomials

1999 ◽  
Vol 101 (1-2) ◽  
pp. 231-236 ◽  
Author(s):  
M Foupouagnigni ◽  
A Ronveaux ◽  
W Koepf
Keyword(s):  
Difference Equation ◽  
Orthogonal Polynomials ◽  
Fourth Order ◽  
Classical Orthogonal Polynomials

scholarly journals Stable Calculation of Krawtchouk Functions from Triplet Relations

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1972
Author(s):  
Albertus C. den Brinker
Keyword(s):  
Difference Equation ◽  
Orthogonal Polynomials ◽  
Recurrence Relation ◽  
Error Propagation ◽  
Orthonormal Basis ◽  
Theoretical Background ◽  
Classical Orthogonal Polynomials ◽  
Krawtchouk Polynomials ◽  
Unit Norm

Deployment of the recurrence relation or difference equation to generate discrete classical orthogonal polynomials is vulnerable to error propagation. This issue is addressed for the case of Krawtchouk functions, i.e., the orthonormal basis derived from the Krawtchouk polynomials. An algorithm is proposed for stable determination of these functions. This is achieved by defining proper initial points for the start of the recursions, balancing the order of the direction in which recursions are executed and adaptively restricting the range over which equations are applied. The adaptation is controlled by a user-specified deviation from unit norm. The theoretical background is given, the algorithmic concept is explained and the effect of controlled accuracy is demonstrated by examples.

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